Special points on intersections of hypersurfaces

We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level $\ell$" points, in the sense introduced by Arnold-Shimura, given as a polynomial in the numbers of hypersurfaces of each degree. Our method builds upon the framework for solvable points of Gómez-Gonzáles-Wolfson to include other classes of accessory irrationality, towards the problem of understanding the arithmetic of "special points." We deduce improved upper bounds on resolvent degree $\operatorname{RD}(n)$ and $\operatorname{RD}(G)$ for the sporadic groups as part of outlining frameworks for incorporating future advances in the theory.